Approximation theory for exponential weights.

UNIVERSITY OF THE WITWATERSRAND

APPROXIMATION

THEORY FOR

EXPONENTIAL WEIGHTS

By

D.G. Kubayi

A THESIS

submitted to the Faculty of Science

of the University of The Witwatersrand

in partial fulfilment of the requirements for

the degree of Master of Science

Degree awa.rded wi~;h distinction on IO December 1998 Johannesburg, Republic of South Africa

Approximation Theory for Exponential

Weights

David Giyani Kubayi

Dept of Mathematics

University of The "Witwatersrand

PO

'Wits 2050

Johannesburg, South Africa

20 June 1998

Declaration

1declare that tills research report is my own, unaided work. It is being submitted for the degree of Mastel' of Science in the University of the Witwatersrand, Johannesburg,

Abstract

Much of weighted polynomial approximation originated with the famous Bernstein qualitative approximation problem of 1910/11. The classical Bern-stein approximation problem seeks conditions on the weight functions \V such that the set of functions {W(x)Xn};;"=l is fundamental in the class of suitably weighted continuous functions on JR, vanishing at infinity. Many

people worked on the problem for at least 40 years. Here we present a

short survey of techniques and methods used to prove Markov and Bernstein inequalities as they underlie much of weighted polynomial approximation. Thereafter, we survey classical techniques used to prove Jackson theorems in the unweighted setting. But first ·westart, by reviewing some elementary facts about orthogonal polynomials and the corresponding weight function

on the real line. Finally we look at one of the processes (If approximation,

the Lagrange interpolation and present the most recent results concerning mean convergence of Lagrange interpolation for Freud and Erdos weights.

To

my father, Mbazima Wilson Kubayi

And

my Dear Wife - to - be Agnes Ngobeni

And

Acknowledgements

I am greatly indebted to my supervisor, Professor D. S. Lubinsky, for suggest-ing the research topic, for his expert guidance and teachsuggest-ing in my research, his invaluable assistance in promoting my knowledge and ability, his encour-agement., patience, enthusiasm and all support I have received from him since 1995.

1.would like to register my sincere gratitude to the Foundation for Re-search and Development for their generous financial support from 1996 to 1995 during my studying for this MSe. Thanks are also dne to the Univer-sity of The Witwatersrand, Johannesburg, for financial support in the form of Postgraduate Merit Award/Postgraduate Bursary from 1996-1998.

Itis my pleasant duty to thank others who have provided support and

guidance. I am particularly grateful to my fellow student, John B. Cyamfi Owusu, for helping me with the typing and the final arrangement of the document. and my friend, Khayizeni Joas Ngoveni, for all the support hn

gave me.

I have tried to give proper credit. to various materials used, but I am sure I havs benefitted directly and indirectly from countless others. lowe many thanks to all the publishers and authr "s concerned.

Imust thank my brother and friend, Mr Jimmy Ngoveni, for his support and his uncountable words of encouragement. which, like streams of water in the desert and the shadow of a great rock in a thirsty land, strengthened me day by day.

Most of all, I am indebted to my parents whose contribution to this effort is truly beyond measure.

D. G, Kubayi

Contents

Abstract .

Acknowledgements iii

1 ORTHOGONAL POLYNOMIALS: PRELIMINARIES 1

L 1 Brief Historical Review . 2

1.2 Elementary Facts . . . . 4

1.3 The Weight Function and Orthogonal Polynomials. 15

2 "WEIGHTED POLYNOMIAL APPROXIMATION ON (-00,00) 25

2.1 In:,rodnction...

2.2 Markov and Bernstein Inequalities.

25

29 2.3 Jackson and Bernstein Theorems For Exponential weights 41

2.3.1 Techniques For Classical Jackson Theorems 2.3.2 Kroo and Szabados' Method .

42

.51

2.3.3 Some Technicalities . . . .. 52

3 L RANGE INTERPOLATION 64

3.1 Introduction... 64

3.2 Mean Convergence of Lagrange Interpolation for Freud Weights 65

3.2.1 Introduction and Main Results

3.2.2 Technical Lemmas ""'"

65

70

3.2.3 Proofs of Theorems in Section 3.2.1 . . . .. 76

3.3 Mean Convergence of Lagrange Interpolation for Erdos Weights 97

Chapter

1

ORTHOGONAL

POLYNOMIALS:

PRELIMIN ARIES

In this chapter, we look at some elementary facts about orthogonal polyno-mials with respect to a general weight function. It is only fitting that before we get. into any technical details, we start with a brief historical review of the subject and its relationship with weighted polynomial approximation.

1.1

Brief Historical Review

The general theory of orthogonal polynomials owes much of its "existence" to the great Russian mathematician Chebyshev. Chebyshev published a series of papers in 1855 in which he discussed integrals of the type

7

P(x) dy, wherep (x) 2:

°

in (-00,00)

-co X - Y

(1.1)

and the sums of the type

00 82

L -'-,

Bi =F0,

x -x·

-00 '

(1.2)

in relation to what was later called the Moment Problem by Stieltjies in his 1894-95 classical paper: "Recherches sur les fractions continues ", Our modern Stieltjies integral comes from this paper.

The Moment Problem: find a bounded non-decreasing function 1p(x) in

the interval [0,(0) such that its "moments" 00

J

xnd1jJ(x), n=0,1,2, .. ·,

o

(1.3)

have a prescibed set of values 00

J

;z;nd1jJ (x)

=

J.ln' n

=

0,1,2,···.

o

Stieltjies borrowed the terminology "Problem of Moments" from Mechanics. Unlike Stieltjies, Chebyshev was not interested in the existence or construe-tion of p.solution of the Moment Problem,

00

J

p(x) xndx =/-Ln) ti=0, 1, 2"", o

(1.5)

but mainly in the following two problems:

(1) How far does a given sequence of moments determine the function p (x) ? In particular, given

00 00

J

p (x) xndx =

J

e-x2xndx, .z =0, 1,2"", (1.6)

-00 -00

can we conclude that

) _ 2

J1(x =e x

or, in our modern terms, that. the distribution characterized by the function

x

J

p (x) di

-00

is a normal one? TIlls is a Iundarnental problem in the theory of probability and in mathematical statistics.

(11) What are the properties of the polynomials Wn (z) ,denominators of

successive approxirr-mts to the continued fraction:

In his attempt to find a solution to (II), Chebyshev opened a vast new field, the theory of orthogonal polynomials, of which at the time only the classical polynomials of Legendre, Jacobi, Abel-Laguere and Laplace-Hermite were known before Chebyshev. In his work there are numerous applications of

01'-thogonal polynomials to interpolation, approximate quadratnres, expansion of functions in series. Later they have been applied to the general them ' of polynomials, theory of best approximants, theory of probability and mathe-matinal stastlstics, And most. recently they have been applied tc the theory of weighted polynomial approximations. This is the area that our investiga-tions will primarily be based. In the sequel we discuss ideas and methods and in between we display some results and their proofs.

1.2

Elementary Facts

Let IF :!R -t [0,00) be 11(Lebesgue) measurable function 011 the real line. The support of ~V.denoted by supp(W) . is defined by

x+e

supp (H') :={:rElR : ',/~ >O.

J

W> 0 }

W is said to be a weight function if supp(W) has a positive (Lebesgue)

measure, and

J

IW

~V(t) dt <00, ti

=

0,1,2, ....

Inthe remainder of this chapter, W denotes a fixed weig.it function, Also we denote the class of all real polynomials of degree at most nby fIn.

Lemma 1Let P be any polynomial which is non -negative on supp(W), and

J

P (x) Hi"(x) dx =O. Then

P (x) =0 \ix E R

Proof. Since P(x) 2': 0 for x ESUpp(W) and P(x) W (x) 2':

°

almost ev-erywhere, it. follows "hat P (x) W (x)

=

0 almost everywhere. Since the set

X+1i

{x ElF.: \ie

>

0,

J

W> o}has positive measure, this implies that P(x) =

°

for all x Effi.. III

Let P,,, Pmbe polynomials. Then we define an inner product, OIl the class

of all polynomials by

(Pn, Pm):=

J

r;

(t)

r;

(t) W (t) dt.

This definition shows that orthogonal polynomials can be constructed using the Gram-Schmidt crthogonalization procedure. This is illustrated in the following theorem:

Theorem 2 There exists a unique (infinite) system of orthogonal (otthonor-mal) polynomials Pn(x) :=Pn

CvV;

x) :=,nXn

+ ...

E Il.,, In :=In (ltV)

>

0,n =0, 1,2,···, (1.7) such that

J

P.(X)Prn (x) W (x)dx = { 1, ifn =m (1.8) 0, ifn:f. m.

Proof. Existence: we write

1

Po (x) := (/ W (t)

dt)

-2.

Having defined polynomials PO,'" ,P~ satisfying (1.7) and (1.8) for some integer k 2::0, we write Uk+l

(x)

k =7'k+l -

IJ

Vk+1,PJ)PJ j=O 1

=

(Uk+l, Uk+l)-2 Uk+l. Pk+1

--- DM7 __

Observe that VI

=

UI. Since the system {xk} is linearly independent, Uk is not identically 0 for any k. Hence Lemma 1 shows that (Uk, Uk) > 0 for all integer k ;::::O. Therefore, the process inductively defines an infinite sequence of polynomials {Pk} satisfying (1.7) and (1.8) Uniqueness: let

be another system of polynomials satisfying (1.7) and (1.8) . We observe that

{Pk}k:J

and {Pk}k:J are both bases for TIn-1 for each integer n ~ 1, and

hence (1.8) implies that

J PpnWdx =J Pp~TVdx

=

0 for all P E TIn-I.

S·_{ince "In}-1_{Pn - In}.-1"_PnETI tl . . n-1, ,_11S gives

o

=- J[I;lPn-I'~-lP~JPnWdX = 1;1J p~1iVdx - "I~-1J P~PnWdX = 1;1 - "I~-l

J

P~PnWdx. Hence,

J.

r

1:-P PnTld:r =-n 'Yn (1.9)

and switching the roles ofPnand p~, we get

J

*rr'd ~/n

PnP"n' X = ::;

In

(l.lO)

Since"t« andf~ are both positive, equations (1.9) and (1.10) imply that 'Yn =f~ , and /P~PnHtdx =1. Then

J

[Pn - p~12vVdx

=

J

P~fV ax _.2

J

Pnp~Wdx

+

J

p~vVdx =1 - 2 (1)

+

1=0

and Lemma 1 implies that

i.e., Pn = p~. III

Remark 1 Given a sequence of positive numbers fn

>

0, we can construct a

'un'iqlLesystem of polynomials orthogonal 'With respect tIl a suitable measure, such that the leading coefficient of the polynomial of degreeri is 't«.

Next, we prove a recurrence relation which underlies t.he importance of

Theorem 3 Let 'Y-l .=0and (3n:=

J

xP~ (x) W (x) dx, n =0, 1,2, .... (1.11) Then (x - (3n) pn (x) =~Pn+l (x)

+

'Yn-l Pn .1(x) ,n

=

0, 1,2, ... , (1.12) ~in+1 "Yn where P-l (x) :=O.

Proof. First, consider the case n =O.Inview of the equations

'Y5

J

vV (x) dx =

J

P6

(x) W (:r)dx =1, and (:30='Y5

J

xTT/(~) dx. we see that

J

(x - (30)W (x) dx

=

J

:clV (x) dx - (30

J

vV(x) dx (30 (30

=

2"-2=0. 'Yo 'Yo

Hence, x-/30

=

~Pl (x), and (1.12) follows for n

=

O. Next, let. 'I 2': 1.Since xp.; (x) EIIn+lo there are numbers ao,Ill,"" an+! such that

n+l

xpn (x)

=

2:

IlkPk (x)

1,,=0

(1.13)

Using the orthogonality relations (1.8), we have

ak=

j

xPn(x)pk(x)W(x)dx, k=O,l. ... ,n+l. (1.14)

We calculate the coefficients ofak one by one. If k :::;n. - 2 then XPJ...(x) E

I1n-1. SinceXPn-1 (x) - "'In-l Pn (x) EI1n_1, we get

In an-l =

j

xPn-I(x) Pn (x) W (x) dr

j [

XPn-t(x) - 1'n-lpn (x) EI1n-1] Pn (x) W (x) dx

+

~/n-l Jr P; (x) W (x) dx "'In "t« 1'n-1 1'n

Similarly,an+1

=

...1!L., and finally,

"'In+!

an

j

XPn (x) Pn (x) W (x) dx

j

xP;' (x) W (x) dx

=

!3n.

(From (1.11)) II

Remark 2 Given a polynomial P EI1n-1, we can write it iii the form

n-1

P (x) =

L.

akPk (x)

k=O

where the constants ak are given by

P(x) :::; ~(Jp(t)Pk(t)I;T/(t)dt)PdX),\lPEl1n-l' (1.16) k=O n-1 J P (t){l: pk(x) pk(t)}W (t) dt, \lP EI1n-1· k=O Therefore, we get

Using Theorem 3,we obtain a simple formula for the important Christoffel-Dosboux kernel

n-1

Kn (x, t) ::::;Kn (Wj x, t) ::::;

l:

pdt) pk(x) . (1.17)

k=O

Theorem 4 (C,'1ristoffel-DarbolLx Formula)

For integer n ~ L

K ( t).- 1'n-1 Pn (x) Pn-l (t) - Pn-1 (X)Pn (t)

n z, .- t'

1'n

x-(1.18)

Proof. Using the recurrence equation (1.12). we get n-1 xKn(x,t) = l:xpk(x)pk(t) k=O :::; ~ {.J!:....Pk+1 (x)

+

f3kPk (x)

+

1'k-1pk_1 (x)} Pk(t) k=O 1'k+1 1'k n l' n-1 =

l:

k-1pk(X)Pk_dt)

+

l:f3kpk(X)Pk(t) k=l 1'k k=O n-1 +

l:1'k-1

pk_1 (X)Pk (t). (1.19) k=O 1'k 11

-Similarly,

n-'-1 . n-1 n

uc;

(x, t) =

L

lk-1pk(x) Pk-1 (t)+

L

tJkpk(X) pk(t)

+

L

1

k-1pk_t(X) pdt)

k=O Ik k=O k=1 Ik

(1.20)

Subtracting (1.20) from (1.19) gives

(x - t) J{n (x,t) =In-1pn (X)Pn-1 (t) - In-1pr. (f)Pn-1 (x).

rt« In

This gives (1.18) . II

Corollary 5 For integer n 2:: 1, n-1

K;(x, x)

=

L

p% (x)

=

']n-1 [p~ (x) Pn-1 (x) - Pn (x) P~-l (x)] (1.21)

k=O /'n

The quantity J{n (x, x) and its reciprocal, known as the Christoffel

func-tion, denoted by An(x), play an important role .'1the theory of orthogonal

polynomials.

Next, we present a specific example to illustrate the general theory dis-cussed thus far.

Example 6 (The Chebyshev Polynomials)

These polynomials are defined on [-1,1] by the formula

Itis well known that 'II'n(cost)is indeed a polynomial of degree nin cost, 'II'(J (x)

=

1 and for integer n ;:::1,

(1.23)

It is also verified easily that

1

J'II'" (x) 'II'm(x) (1- x2)-~dx

=

~,

ifti

=

m

=J

0 (1.24)

-1

0, if n

=J

m

7r,ifn

=

m

=

0

Therefore, the system

{If.

(If)

'II'1(X),{g'II'2(X), ... }

is a system of orthonormal polynomials with respect to the weight function

{ (1- x2)-~ , if

Ixl

s

1

TF(x) ==

o

,

if

Ixi

>

1

(1.25)

The trigonometric identity

2cost cosnt =cos(n -+- l)t

+

cos(n - l)t

takes the form of the recurrence relation

which is consistent with (1.23) and (1.24). The Christoffel function An (x) can be calculated explicitly. By definition,

so that ~ + cos2t+ ... + cos2(n ~ 1)t 2 ~ + (1+ cos2t) + ... (1+cos2(n -l)t) 2 2 2 n 1~ - + ~ LJcoskt. 2 2k=l

Multiplying both sides of the above equation by 2 sint,we get

n-l

'lfSintA~l(cost) nsint+ 'L:sint cos2lct k=l

1n-1

= nsint +:2

E

[sin(2k

+

l)t - sin(2k - l)t]

. 1. (')) 1.

= nSIDt

+ -

SlIl ~n - 1t - ?SlIlt.

2 ~

Hence,

-1 1 1 sin(2n - l)t

An (cost) =-{n - -2 + 2' t }.

'If SlIl (1.26)

It.can be verified directly that

sin(2n - l)t

is a polynomial in cost of degree 2n - 2. ThL, polynomial is U2n_2.Thus,

.ed by

(1.27)

In the next section, we define the class of weight functions, known as Freud weights, and the corresponding orthogonal polynomials. vVe also study some simple properties of these weight functions and polynomials.

1.3

The Weight Function and Orthogonal

Poly-nomials

In this section, we introduce Freud weights and study some of their ele-mentary properties. We will also estimate the order of magnitude of the recurrence coefficients In and the largest zeros of the ort.hogonal polynomials with resp -ct to these weights.

fu;~cf,ivn , if

Q (x) :=log {W~x)} (1.28)

is an even. convex function on JR, Q is twice continuously differentiable on

(0,00) and there are constants Cl and C2 such that

xQ" (x)

0< Cl< Q' (x) < C2< 00, "i/x E (0,00). (1.29)

We will often write WQ (x) := exp (-Q (x)). In the remainder of this section, vV

=

WQwill denote a fixed Freud weight function, The prototypical Freud weight functions, exp

(-lxI

O), Cl:

>

1, will be denoted by Wo' The other

cases, 0

<

ex

<

1 and Q :::::1, will be dealt with in chapter 2.

I

,

Notation: A "" B means that, clA::; B ::; c2A. Also the integral

without limits denotes the integral over the whole real line.

'vVe enumerate some simple properties of Freud weights in the following

proposition.

Pror-ositlon 8 (n).'f 0

<

;C

<

J/

<

00 then (with constants ell ('2 us in

(1.29)),

•

7!F_ sy=r==_ =_T _

J

Ixl

nW (x) dx < 00, n

=

0,1,2, ... , (1.32)

In parUcular,

lim Q' (x) =00.

x-too

(b) The weight function W satisfies

In particular,

and

lim P (x) W(x) =0

Ixl~oo

(1.33)

for every polynomial P.

(c) For x '2:Q' (0+), let qx denote the least po.ntive solution of the equation

Then

Remark 3 : The numbers qx introduced in (1.34) nrc called the Freiul. num-bel'S. VVe obseroe that

if

IV is a Freud weight, so is n:2. Proposition 8 and

results in section 1.2show that there exists a system of polynomials, called Freud polynomials,

Pn(x) :=Pit

(nf2;

x)

:=,nXn

+ ...

E TIn, In> O. (1.35) such that

{I, ifn =m

J

Pn(x) Pm(;1;) W2 (.c) dx = .

0, ifn:f m

(1.36)

Moreover, for each n ~ 1,Pithas n real and simple zeros,

(1.37) We note that the zeros are .symmetric around the origin, since the weight function is even. The recurrence relation given in Theorem 3takes the .form

() In-l () l'l-2 () ')3

XPn-l X

=

=r=p« :1;

+

--Pn-2 X , n

=~, ,....

In In-l

(1.38)

In this section, we estimate the order of magnitude of the recurrence coefficients ~'''-l and the largest zero Xn.

"In

First, we make some observations concerning orthogonal polynomials.

Proposition 9 We have 111-1 =

J

XPn(x) Pn-1 (x) H,2(x) dx. In ~ In-l

=

I

Q' (.,.) Pn(;1;)Pn-l (.r.) H'~(X) dx, 2 "t« . n

+

§

=

J

xQ' (x) p~(x) TV:! (x) dx. (1.39) (1.40) (1.41)

Proof. Since XPn-l (x) = 7"-lPn (x)

+

P(x) for some P E TIn-I,

rt«

then equation (1.39) follows from the orthogonality relations (1.36). Next, we observe that p~is a polynomial of degree n. - 1with leading coefficient n'Yn. So, p~ - n...:1!L.pn_1 E TIn-2• Using the orthogonality relations (1.36)

"1',,-1 again, we get

=

n~

Jp~-l

(x) W2 (x) dx 'Yn-1 'Yn = n--. 'Yn-1 Integrating by parts and observing that

lim P~(x) Pn-1 (x) W2 (x) =0,

Ixl-oo

we deduce that

n~ =2

J

ci

tx) Pn(x) Pn-1 (x)

vV

2 (:1:)dx -

J

Pn (x) P~-l (;7:) W2 (x) dx,

'Yn-·1

Since P~-l ETIn-2,the second integral above is zero, and we get (1.40) . In order to prove (1.41), we observe that xp~ (x) - tip.,(x) E TIn-1, and hence,

Therefore, an integration by parts as before,

11

+ ~

= ~

J

[2xp~ (x) Pn(x)

+

p~ (x)] W2 (x) dx

= ~

J

:X

[xp;, (x)] W

2

(.7;)dx

=

J

xp;, (X) Q' (X) W2 (X) dx which is (1.41) .II.

\Ve are now in a position to estimate the recurrence coefficients '1,,-1_"In and the largest zero Xn• These estimates are not sharp for Freud weight functions.

Theorem 10 We have 1 In-1 ') 2 3 -qn ::;-- ::;-'In' n= , ,... , 4 111 (1.42) and (1.43) In particular, 111-1 v -- rv ./\.11rv q". 111 (1.44) Proof. Inview of (1.39) , ~,n-1 = {

J

+

J

}XPn(x) Pn-1 (:1:)

n'

z (x) dx, In Ix::O;qn Ixl?<Jn (1.45)

Using Schwarz inequality, we get

I

\ J

XPn(x)Pn_r(X)W2(;l::)dx

S s« / IPn (x) Pn-dx) W2(x)1 d:t

I I

S qn(/p;(x)W2(X)dx)2 (/P;_1(X)H:2(x)dX)2

( 1.46)

Since W is even, and Q' is lncreasinr , we have

/ XPn (X)Pn-l (x) W2(x) dx Ixl~qn ~ 2llxp• (x)

P._,

(x) '1" (x) -2 qo

<

Q' (qn)

L

IxQ' (x) Pn (x) Pn-l (x) W2 (x)1 dx ') 00 S Q' (qn)

l

lxQ' (x)Pn. (J:)Pn-l (x)

n·

2(x)1 dx = ~~/lxQ'(X)Pn (x)Pn_dx)TV2 (x)ldx (1.47)

We observe that xQ' (x) H/2(x) 2:: 0 for all x E lR. Hence, using Schwarz inequality and (1.41), we get

/ IxQ' (x)Pa (x) Pn-l (x) VI/2(x)1 dx I I S (/ XQ'(X)P;(x)TV2(x)dXr (/ XQ'(x)p~_dX)W2(X)d;L')2 =

_V'

rn+~/n-~=/n2_~<n.

(l.4S) 2 2 4 -21

Substituting from this estimate into (1.47), we get

J

XPn (x) p,,-dx) vV2 (x) dx 5:.qn

jxj;:::q"

(1.49)

Using the estimates (1.46) and (1.49) in (1.45) , we get the upper estimate for the recurrence coefficients in (1.42) . For the lower estimate, we write (1.40) in the form

~1'::1

={

J

+

J

}Q'(x)p" (x)Pn-dx) W2 (x)dx. (1.50)

jxj:O:;qn 142:qn

SinceltV is even, and Q' is increasing, we use the Schwarz inequality to get

Similarly, using the fact that W is even and the estimate (1.48), we get

J

Q'(:.~)p,,(x)Pn_t(x)W2(x)dx Ixl;:::qn

< ~

J

xQ' (x) IPn (x) Pn-l (x)1vV2 (x) dx qn Ixl;:::qn (1.52) n

<

Frein (1.50), (1.51) and (1.52), we get,

n 1'n 2n

---<-21'n-1 - q,,'

and hence, the lower estimate in (1.42) . To obtain the bounds on the largest zerosXn, we can use the relation between the recurrence coefficients and the

zeros to show that

X

Ik-l

n<2 max -- < 2 max (2qn) S4qn-l,

- lSk$.n-l Ik - lS:k$.n-l

which is the upper estimate in (1.43) . Similarly,

X > Ik-l > In-2 > qn-l

max -- -- --.

n - lS:k$.n-l Ik - In-l - 4

Finally, we obtain an upper bound on the leading coefficient itself.

Proposition 11 For integer n ~ 1,

In Sc (12)n

I]n

Proof. From (1.42),'we get

In ,,,-1 Ii In

=,0 ----

...

-In-l/n-2 'YO 4n ( )-1 S 10

s«:

ql Since n n k

log n!=

2:

logk ~

L

J

logxd» =n log n - n

+

1.

1<=2 k",,2 k-1

(1.53)

we have

nn

_ < n-l < 3n-1

1- e _ .

n.

Chapter 2

WEIGHTED

POLYNOMIAL

APPROXIMATION

ON

(-00,00)

2.1

Introduction

Most of the great work in weighted approximation of this century originated with the famous Bernstein's problem in about 1910/1911.. We know that polynomials are unbounded in unbounded sets. So Bernstein realized that there was a need to introduce an extra condition on them so that. they can

be used to approximate. He introduced a weight function 'VV that will be

used to weight these polynomials.

Bernstein's Problem: let ltV ; ~ ...(0,1] be continuous. Under what conditions on 'VV is it true that for all continuous

f ; ~ ...

~

and for e

>

0, we can find a polynomial P such that.

IIU -

P)

WIIL",,(:R)

<e

t

We observe that for the problem to make sense, we need

IIPWIIL""

to be finite for every plynomial P, and hence we need

lim xTtW(x) =0, n=0,1,2, ....

10;1-00 (2.1)

Then PW vanishes at

±oo

for each polynomial .P,so we need lim f(x)W(x) =0.

10;1-00 (2.2)

We also need

f

to becontinuous, as the locally uniform limit of polynomials

is continuous,

With (2.1) and (2.2)as above, we will say that the polynomials are dense

Bernstein's approximation problem WEtS solved in the 1950's indepen-dently by Achieser(1954) , Pollarcl(1953) and Mergelyan(1956) . All three so-lations were different , but all involved "regularizations" or funetionals of the weight and are implicit. Mert, yan's solution, for example, involves the function

n

(x) :=sup

{IP

(xjl :

P

a. polynomial with

II

(~~V1~:)

to{R) ::;

l}

He showed that Bernstein '8problem has a positive solution if and only if

J""

Inn (x) dx =00. 1+x2

+oo

In our discussions, we arb going to restrict ourselves to the methods which im-pose regularity conditions on the weight function VVthat guarantee a simple solution. The following theorem '.J,!ne to L. Carleson and M,M. Dzrbasjan:

Theorem 12 Let W = e-Q where Q :lR-+[0,00) is even, raidQ(eX) is convex on (0,00). Then Bernstein's problem he ~ a positive solution, tha:t is the p(Jlynomials are dense for VV, iff

J""

Q (x) dx =00, 1+x2

o

(2.3)

I

z

-=p 44777

Corollary 13 Lf't 0:>0, and

w{~

(x)

.=

exp

(-Ixl")

(2.4)

Then the polynomwls are dense for'

~r

iff Ct '2 1.

With Bernstein's qua)' ,,<.tiveapproximation problem solved, clearly the

next, step for researchers/approximators is the quantitative approximation:

how large must the degree ofP be, to achieve a desired accuracy e '!

Dzr-hasjan and his co-workers had already st.arted investigations in this direction

even before the full solution to Bernstpin's problem was published. The

search for the quantitative sclutlon led to the beginIlillg of the main theme in weighted polynomial approximation of this cent.ury: characterizing the rate of decay of polynomial approximn.tion of a function ill terms of its

struc-tural /nmoothness properties.

Before we get. deeper into these results, we present ideas and methods used ill proving Markov and Bernstein ineqnalities.

----m---I .... RB ..ETI..f'... H*..F..F ..mR ... 4~se...•

2.2

Markov and Bernstein Inequalities

In the sequel,Pn denotes an algebraic polynomial of degree S:: n.and C, Cl,C2, •••

denote positive constants independent; (.Jn, Pn and x. The same constant

does not necessarily denote the same constant in different occurrences. vVe

write C

::f

C(I) to indicate that C is independent of fane! C =C(f) to

show that it depends on

f .

The focus here is on weighted analogues on (-00,00) of the Markov

inequality

ilp~lI£l'[-l'l]:::;Cn21IPnIILp[_1,1]

and the Bernstein inequality

(2.5)

(2.6)

where C

=

C(n,p).

For the canonical Freud weights H~., the Markov inequality has the form 711-~, ifa ~ 1

Ilp~Wc<III,p[a]:::;

CIIPnHTnI!Lp[:R] x logn, if 0:=1 (2.7)

1, ifn< 1

Here 0 <p :::;00 and C

=

C(~1:,p).

We shall dwell at length on the techniques used to prove Markov and Bernstein inequalities, as they underlie much of the weighted approximation

in general.

(I) Dzrbasjan's method [13]

Let €

>

0 and x E1'\. From Cauchy's integral formula for derivatives,

Ip~

(x)\ = 1

J

Pn(t).,dt

27ri {t:!t-x!=s} (t -

xt

1

< ~

!lPnIlL",,({t:lt-xl=e}) (2.8)

This gives as an estimate for the derivative in terms of values of Pn in the complex plane. To return to the real line, we use an inequality that arises in

the theory of functions analytic in the upper-half plane: . y ooJ log!Pn(s)!

log!P,,(x+zy)!::::; - )~ .,ds, y:> O.

1f (x - S

+

y~

-00

(2.9)

To estimate the RES in terms of

we split the integral in (2.9) into several pieces. If;1:

>

8

>

0, and Q iH

x+fi

Y

J

d«

:::; log AI

+

Q(;1

+

8) - 2 ').

7[' (:l:-J)+y~

x-a

The integral over the complementary range is estimated using the

Bernstein-type inequality:

where Tn(x) denotes the Chebyshev polynomial, Pn is an arbitrary

polyno-mial and

Ixl

>

1.

A careful choice of a small value of y then yields the required result.

(II) Freud's Method I [16], [17)

There are two methods to consider here. We begin with the one for the general case. Assume that

n'

:=e-Q, where

q

is even, convex and of smooth

polynomial growth at.00. Let {P.1(x)}~o denote the orthonormal polynomials

with respect to W2 so that

ex:

J

pj (x) Pk (x) ~V2 (x) dx =OJ!,. (2.10)

-00

For

f :

lR->IR such that,

f

(;1:)W2 (x) ;;.;i ELI (IR),j ;:::0, we can define the

formal orthonormal expansion

oc oc

t

r- 'LbJPj, bj

=

J

fpjW2,

.>

O.

i=» -»0

(2.11)

Let

m-l

s;

[fl

:=

L

bjpj j=O

(2.12)

denote the mth partial sam of this orthonormal expansion. Using ideas of T. Carleman from Fourier series, and local one-bided approximations to VV,

Freud proved the (C,1) -bound

(2.13)

with C

=f

C (f, m). Here am = am(Q). The class of weights included

vV',..,

0: 2: 2, for which :.:.

=

Clm1-i. Once we have this, the sup ..norm

Markov inequality follows. For, introducing the dilated de la Vallee Poussin

S11m 1 2m Vm[f]= -

L

53

[fl

Tnj=m+l (2.14) we see that (2.15)

Also Vmhas the quasi-projection prop' ty

duality and (C,l)~bounds such as

II~

t

ISj

[fJI

wll

~

C3

1If

W

II

L",,(:R)

)-1 L",,(R)

to pass to an Lt analogue of (2.15), and interpolation gives it for 1<p < 00.

(2.16)

(III) Freud's Method II, Levln-Lubinsky's Method [24], [25J

We illustrate the idea. for

rv

:= e-QI as used by A.L. Levin and D.S.

Lubinsky. Suppose that we can find J( > 0 and polynomials Sn of degree

~ Kn. such that for

Ixl ~

2an!

(2.17)

and

(2.18) Then using their infinite-finite range inequality:

Levin and Lubinsky showed that

I

I

P~WIILp(lR) ~

IIP'WII

n Lp(-an,an)

~ Cc

1

1

II

(PnSn)' - PnS~11 . (2.10)

Lp(-tl'l'Cln,)

Now by the classical Bernstein inequality (2.6) for [-1, 1Jscaled to [-2an, 2an

l.

we have

Using this and (2.18) in (2.19) gives

(2.20)

(IV) Nevai-Totik's Method [55]

We assume that W

=

e-Q, where Q is concave and Q(0) =O. Here we

make nse heavily of the concavity and of fast decreasing polynomials. Let us suppose that forn 2:: 1, and some K > 0, we have polynomials 'or;:of degree

N =N (n) say, that decrease rapidly in [-1,1] in the sense that

ISn (x)1 :::;I<e-Q(anx) , x E[-1,11· (2.21)

but

Sn

peaks at 0:

Sn (0) =1;

5:.

(0) =O. (2.22)

By a translation, we obtain polynomials Sn of degreeN (n), such that

(2.23)

and

S;(0) ==1; S~(0) ==O. Then for the Markov inequality at 0,

I(P~W) (0)1 = IP~(0)1

= \

(PnSn)' (0)1

:s;

n

+:

(n) IlPnSnIILe<>[-an,an]

n

<

J{n.

+

N (n) IlPnWIILco(lR) . (2.24)

an

Here we have first used t.he classical sup-norm Bernst.ein inequality for [-1,1] sealed r\ [-an, an], and then (2.23) . For general x ~ 0, we use the evenness

and concavity ofQ t.o deduce that.

W (x)

w

(y)

s

W(x

+

y).

Now fix x ~ O. Applying (2.24) to the polynomial

R"

(y)

.=

r:

(x

+

y)

tv

(x) gives IP~WI (x) == IR;,(0)W (0)1 (W (0) =1) < J{n+N(n.)IlRnWII an Loo(RJ __ J{n+N(n) supIPn(x+y)IH'(x)W(y) an yeR < J(n+N(n)IIPnWII an L",(Rl 35

What are about the size ofN (n)? For W

=

Wi, so Q (x)

=

[z],and an

=

~n, Nevai and Totik showed that we can choose N(n) =0 (nlogn) ,so that

If on the other hand Qis concave, and

00 Q (x)

j

-1--"dx <00

+x-o

then we can choose N (n) =0(an), and an grows faster than n, so

IIP~WII

::::;

c

IIP"WII

Loo(l!)· Leo(l!)·

To obtain the fast decreasing polynomials {Sn} of minimal degree, N evai and Totik used a construction of Marchenko on entire functions of exponential type.

(V) Levin-Lubinsky's Method [26J , [27J

Initially unaware of Drzbasjan's work, Levin and Lubinsky used Cauchy's integral formula for derivatives to derive (2.7):

IIP~WCtIl

:5

C

IIPnWIl

x logn, if a=1

Loo(nI) Leo(a)

1, if a

<

1

However, rather than using (2.9), they used the weighted Bernst.ein inequality:

IP

n

(z) G (z)1 ::;

l\PnWIILoo

(-a,a)' z E

C\

[-a,a1

to est.imat.e IPn(t)1 for It - xl =s ;i.e.,they used

Here t.he explicit representation

(where p" (,) is a non-negal"'" "de",ity" function and a" is a con,tant)

plays a role. By estimat.ing the density function iJn (s) and t.hen the func-t.ion \Gn,aIL (t)r1, and carefully choosing c, Levin r,ud 11tbinsky proved the

following t.heorem:

Theorem 14 Let W :=e-Q, where Q :1PL_" 1PLill even, Q(f))

=

0, Q" is

contin'uous in (0,00) , Q'

>

0 there, and for some A, B

>

0,

(2,25)

Let Q[-l) denote the inverse of Q on (0,00). Then

\\P~W\lL.(')

:5

C

G

Qt!l~(')}

\\P"IV\\,_",'

(2.26) The condition (2.25) is fairly typical of those used in the theory of Freud

Levin and Lubinsky in [26} were not interested in Markov inequality (2.26) ,but in its sharper Bernstein cousin. On [-1.1], the Bernstein in-equality offers its best estimate for P~(x) at x ==O. For Freud weights, the Bernstein improvement occurs not at 0, but at the effective endpoints ±an.

To describe this improvement, they used

1

¢n (x) :=max

{1-

El,

n-~}2 :::;l.

an ('!.27)

Theorem 15 Assume the hypotheses of Theorem 14with A >1. Then

In particular,

2

j(PnW)' (±an)j :::;C71,3 IIPnW!I .

an Loo(:;() (2.29)

The case Q(x) ==

Ix I" , a:::;

1, is excluded by the condition A> 1. When

A :::;1, the conclusion of (2.28) holds for

Ixl ;::

Wn, for any fixed e

>

O. It cannot hold inside (-can, can) in general, as

a:

-+ 0 asn-+ 00, whereas the

correct Markov factor in (2.26) is bounded below by a positive constant.

Kroo and Szabados rediscovered and refined Drzbasjan's method, but they also had the infinite-finite range inequality. They weakened the

condi-tions on Q of Theorem 14 showing that. we need something like (2.25), only

for large x, and we could replace the lower bound in (2.25) by

XQ' (x)

Q(x) 2:A > 0 for largex.

The first Markov inequality for Erdos weights appeared only in 1990[32]. One of the reasons for this delay was the difficulty in fornn lating hypothesis:

how does one describe smooth but rapid growth ofQ ? D.S. Lubinsky [32]

found it convenient to use

XQ" (x)

T(x) =:1+ Q/(X) , (2.30)

though for most purposes, we can use

-T ( .) ._ XQ' (x)

X .- Q (x) , (2.31)

which involves lower derivatives, and typically grows at the same rate as 'T.

Lubinsky then proved the following Markov-type inequalit; :

Theorem 16 Let W ;=e-Q be even, letQ" be continuous in (0,00), Q'

>

0

there, ?I)ithpositive right limit at0, and limit 00 at 00, and

nNI _rssc

_see_

(2.32)

Then

The Lp analogues were subsequently proved by D.S. Lubinsky and T.Z. Mthernbu [42], [43], under a few more restrictions on Q. An interesting featnre here is that Q' (an) grows essentially faster than .!'-,nr, the Markov factor for weights such as W", , ~ > 1,and may even grow faster than n. For example, if Q(x) :=Qk,,,,(x) =pXPk(I:rl"') then [k

]1

Q'(x) rv n (Iog,

nr~

II

log)n. ),,=1

Here exp, :=exp(exp( ... exp(lxl"'))) (the jth exponential), and log, is defined in the same way, Note that for ~

>

2, Q'(an) does grow faster than n.One form of the Bernstein inequality for these weights is [43] :

So we reduce Q'(an) 1;0 .!!.. by inserting the root factor in the LHS. However,

an

the "true" Bernstein inequality involves (r

,Tn'

(x) and is quite complicated: There are different phenomena for three ranges of:1:in [-a'l> CIl]' rather than

40

---two as in the Freud case or the classical (-1,1] case; namely near 0 and near

±1.

2.3

Jackson and Bernstein Theorems For

Ex-ponential weights

In this section we shall cliseuss some of the results concerning the ('ho.1'a('-terization of the tate of decay of polynomial approximo.tion of a function in terms of its strndnral/smoothness properties. In partkular, we shall discuSS

Jackson type estimates for

En(flp :=

\\(f -

P) W\\Lp(Il (2.33)

where I=(-1,1) or~, 0

<

p ::; co and W

.=

e-Q is an exponential weight.

As we have seen in the previons section, there are two moin elasses of

weights on JR.:

e The Frued weights, where Q is evc'n and of polynomial growth at co. The prototypical Frend weights are the W" that we have already seen .

.. The Erdos weights. where Q is even and of fast!'r than polynomial

---I

,

growth at 00. The archetypal examples of Erdos weights are

Here (k

>

0, k ;::1 and eXPk:=cxp] exp( ... exp(lxl") is the kth iterated

exponential.We also set expo (x) =x.

One of the main problems in proving Jackson theorems for exponential weights is the rapid decay of the weight at ±1 or ±oo. So the techniques for proving Jackson theorems are a key factor. We shall concentrate on the techniques in our discussions.

2.3.1

Techniques For Classical Jackson Theorems

In this section we shall outline five methods that have been, or can be, used to

prove Jackson theorems in the classical unweighted setting of C[-l,

IJ -

the

continuous real valued functions on [-1,1] with uniform nr"Ill;or C2,,-the continnous 2'iT-periodie functions on [0,2rr]with uniform norm; or inL1[-1,1].

Our main objective is to review methods that. are very popular, or might have some applications to weighted approximation.

(I) Convolution

I

2rr

J

If" (u) du

=

1.

o

This is the most popular method of approximation when it, comes to Jack-son theorems. Even 'Weierstrass' 1881 proof of his approximation theorem

was based on convolution with the scaled heat kernel exp

(_~)2,

8 > 0.

Jackson then modified Fejer's kernel

1 [sin.( ~~)

ul:!

F,,(u):=,)( _{~ n+} 1) (

sm

2)

to obtain the Jackson kernel

[sin

(Zlf)l4

Kn (u) :=Cn sin (~) , u E [0,271']

where O«is chosen so that

Hence we obtain the Jackson operator

2rr

In [f](8) :=

J

f

(t)

tc;

(8 - t) dt, 8E [0,27r] ,

f

E C2rr. o

Jackson then used In[f] to approximate functions satisfying a Lipschitz eon-dition of order 1, and then via a piecewise linear approximation to general functions obtained his celebrated estimate for

(2.34)

(whereT«denotes a trigonomet.rie polynomial of degree:S71), namely

whereC

#-

C(f,71) and w denotes the usualmoclulns of continuity given by

w(f; 8):= inf

If

(x) -

f

(y)l.

!x-!J!::;6

(II) Fourier Series, Multipliers

This is probably the seconclmost, popular approach to Jackson theorems.

If f EGl11"jand

is its Fourier series, then the partial SUIllS

n

Sn[f1 (x) := ~.

+

'E

(aj cosjx

+

bj sinjx) ~ j=l

are naturally best. hpproximants in the L2 norm, but. in the Loo norm, may

incur a projeetion factor of order l1g11.when compared to best, approximants. The next step then is to modify the partiel sums. This idea goes back to

Cesaro meane ot the partial sums, and Lipot. Fejer's theorem that the averages of the first Tl partial sums of the fourier series off EC211"converge uniformly

j +1

Aj := - (-1)l Cl:.j

2

"multipliers" by solving a dual LJ approximation problem. Favard formed the linear operator

n

L; [f] (e) :=~o+

2.:

Aj (aj cosje +bjsinje) (2.35) J=l

where the {Aj} are chosen according to the rule

and the {Cr.j} solve a certain dual L1 m' 'mization problem, namely

rr

I

n

I

2

I

x -

E

Cl:.ksinkx dx =min =2

(~"l'+1)'

(2.36)

An integration by parts and some manipulations give the identity

u;

[f]- f) (e) =2~

2

[t -

f:i

Cl:.kSinkt]

l'

(e

+

IT - t) dt (2.37)

and hence the 1937 Favard's estimate

En [1] ::;

IILn

[.f]-

fII

L",,[O,2rrj

~ 2(n7~ 1)

1I1'IIL",[O.2~1·

(2.~8)

The constant is sharp for each 11 in the class of functions with continuous

derivatives.

\

_~

i

(III) The "Unknown Approximator's" Method

The name of t.hiSmethod is due to D.S. Lubinsky. The origins of this

method Rrr -t known; though it may have been used by Dzadyk. Ron de

Vore used it in 1968. The method involves several steps. Let

f ;

(-1, 1}-+ lPt. Step 1.Approximate locally, ouer subintervals of (-1,1].

Choose a partition of (-1,1], not necessarily uniform,

-1 =Yo

<

Y1

<

yz

< ... <

Yn

<

Yn+l = 1 (2.39)

and on ('}1 'lpproximate

f

by a polynomial of low degree, let us say, a

cubic polynomial. This giveHthe approximation n

.f

:Y. I:P{:((YJ,Yj+l) j=O n-1

=

Po

+

I:(Pj+1 - Pj)

X[Uj,YJ+t) j=o (2.40)

where XA denotes the characteristic func.tion of a set A. Step 2. Replace X[yj+I,l) by a polynomial Rj.

Now approximate each characterist.ic function X[YJ+I,l) ina.suitable sense by a polynomial Rj of degrea ::; n, to give a polynomial approximat.ion

n-1

f ~

Tlo

+

I:

(pj+1 .- pj) Rj+1 =;p

)=0

(2.41)

of say degree ::; n

+

3. This method has been extraordinarily useful in

This method was first formulated in full in B. Bojanov's 1995 paper [2}. preserving), approximation and also worked well in Lp spaces, 0

<

p

<

1, where other methods fail.

(rv) Bojanov's Method

The method is based on the alternation theorem: ifp~is a best. polynomial

approximation otdegree S n to

f

in the uniform norm on [-1, 1},so that

then there are at least.n

+

2 alternat.ion points

-1 :::; No<.!Jl < ... <y".,..l S1

such that

(f-P~)(YJ)=±(-W En[f}, O:::;j:::;n+l. (2.43)

From this one can derive an explicit formula for E"

[f1

in terms of the deter-minants that arise in studying Chebyshev systems: Define

n En [f] =

L

(-1)"

c"

[j (Uk+!) - j (yk)] k=O (2.45) Then

[1

z Xn

y:+<

1

Yo Yt Yn E.

[fl ~

[1

.

1

(2.44) x xn Yo Y1 Yn

whereoS is any function with S(Yj) =±(_l)i. A little manipulation of the

determinants leads to

where the coefficients{Ck} all have the same sign and satisfy

(2.46) One readily concludes that;

(2.47) But we can get much more information by estimating the sum

n

L

Ic,,1

(Yk+1 - Yk) ,

k=O

which turns out to be related to the L1 norm of a certain piecewise constant, function. This method at present yields only

En [f]

s

CW

(j;

)n) ,

(2.48)

_e_n

&2

p= Fa&&!

En

[/1 ::;

Cw (/;~) .

weaker than the Jackson rate. This is promising and one day it will surely lead to the Jackson rate:

(V) The L1 Christoffel Method.

This method was first applied by 11. Rlesz in his classical theorem

C0I1-necting the poss 'ty of one-sided L1 approximation to the uniqueness of

the solution of the Harmburger Moment problem. As such it fits more within the framework of weighted approximation on unbounded intervals. Here we present its possibilities for(-1,1) .

The nth Christoffel function associated with a non-negative measure df.L

on the real line is definedby

\ ( ) •• J:

J

P'~-l(t) d ()

I\n du, x := mr p2 () f.L t .

Pn-l n-1 X

(2.49)

Let

r~

(x) denote the step function

rd.)

:= {

1,

x:::; ~

(2.50)

0, x

> ~

Then one can construct <Pn, Witof degree:::;2n - 1, such that. in ffi.

(2.51)

-We obtain <Pn and Wn by Hermite interpolation of

r~

and its derivatives at Gauss type quadrature points that include F. Moreover,

In particular, then

The point of this method is that for a great deal of "nice" measures u, one has the correct estimates for the Christoffel functions I\n (dfJ" x) and hence a good estimate for the LI error of polynomial approximation of step functions. One can extend this to fairly general functions by first approximating them by splines or one can use Bohr type inequalites and approximate functions of

boundr variation. This idea can be extended to general Markc : jChebyshev

systems.

Next we present two methods that are used to prove Jackson theorems ill the weighted setting. The methods used to prove Jackson theorems in weighted approximation on lR were baser on modifying orthogonal expan-sions and the LI Christoffel method. Round about 1993, two sets of authors begar to consider alternative methods: on the one hand, Kroo au.I Szabados

developed Bojanov's method and on the other hand. Ditzian and Lubinsky investigated the possibility of applying the unknown approximator's method. We :tart with Kroo and Szabados method.

2.3.2 Kroo and Szabados' Method

Let

I :

lR -? lR be continuous with

IW

vanishing at

±co.

Let p~be a best

polynomial approximation of degree :5nto

I

in the weighted uniform norm .. Then there exist at least n+2 alte.riation 'nts

Yo

<

Yl

<

Y2

< ... <

Vn+l with

[(I - P;) W] (Yj)

= -'-

,-l)i En[f]""" 0:5j :5n.

+

1. (2.53)

Kr06 and Szabados [23] showed

a

Iii.Bojanov, that

z,

[flo<> =

I't

C-1)k dk

[I

(Uk+l) -

I

(YIc)]

I

k=O

(2,54)

where all dk > 0and

:t

dk [W-1 (Yk+t)

+

W-'Y (Yk)]

=

1.

k=O

(2,55)

However, the real art lies in their use and estimation of

en

;=

i:

dk

[n--'Y

(Yk+t) -: H -'"((Uk)

1

(Uk+l -. /}i) . (2.56)

Here, E(0,1) is fixed. If Weintroduce the modulus

(2.57)

Kro6 and Szabados showed that

(2.58)

For a large class of weights, including ~Va,co> 1, they proved that if all

8_{n_}-::cqnlog~

n (2.59)

and hence deduced the estimate

(2.60)

The exponentially decaying term reflects the inability of weighted

polynomi-als to approximate outside [-Cqn' Cqn]'

2.3.3

Some Technicalities

Before we turn to the unknown approximator's method in a weighted setting,

we need to present a taste ofS01118 fundamental advances made possible by

the systematic application of potential theory in the 19S0's and 1990's.

(2.61)

\Ve concentrate on theform of Freud's infinite-finite range inequality due to Mhaskar and Saff, Let I =(-1, 1) or JR., and W :=e-Q, where Q :I -->lR is even and convex in I and hae limit 00at the endpoints ofI. Then Mhaskar and S;>ffshowed that ifan> 0 is the root of the equation

then

(2.62)

and that asymptotically as n.--> 00, an is the smallest such nnrn'<er, Later subsequent discussions allowed the convexity of Q tobe replaced by xQ' (x) being increasing on In (0,00) .

In the Lp case, \ve cannot hope for equality of the Lp norm over'I with that over some subinterval, as we are missing out part of the integral. So the Lp analogue of (2.62) has the form

where 0

<

p

<

00, s >0 and C ==C(6') .The last inequality in (2.63) follows from the inequality

(2.6·1)

Until recently, almost all the work on weighted approximation on ~

concen-trated on the so-calledFreud case, where Q is of polynomial growth at 00.

This is due to the fact that handling the growth ofQ and describing

conse-quent behaviour is more difficult when Qis of faster than polynomial growth

XQ" (x)

T (x) :=1

+

Q'

(x)-at 00,

In an at'iempt to find a function that; could guarantee the regular growth ofQ, D.S, Lubinsky [32] used the function (2.30) :

though for most purposes, we can use (2.31) :

- xQ' (x) d

T(x):= Q (x) =xd)log Q (x)].

Both quantities typically grow at the same rate and it is not difficult to see

that if T (x) is bounded as x --+ 00, then Q(x) is of at most polynomial

growth at 00, while ifT(x) --+ 00 as x --+ 00, then Q (x) is of faster than

polynomial growth at 00.

'vVepresent three examples to illustrate the growth ofT and a71•

Let W (z) ::::;W"(x) ::::;e-1xl". We see that T(x)

=

x [-ax,,-lj =a -x" and which gives where

Example 18 .Some Erdos Weights

LetW (x) ::::;T%,,, (x) ::::;

exp C-e~IQ)

,where eXPk denotes thekthiterated

exponential. 'Ve see that, after some manipulations, k-l

T(x)

=

C'i!X"

II

exp, (x"), x> O.

j=l

An application of the rudiments of Laplace's method to the integral defining an gives

{ ( lk+l )}~

an::::; logk_l logn -

2" ];

log}ti

+

0 (1)

and hence

Here logj denotes the jth iterated logarithm. One can also show that

k

T(an) '"

II

log,77..

j=L

Example 19 Some Exponential Weights en (-1,1),

Let W (x) =wO,a (x) =exp (- (1 -

x2r") .

Then we see that

and

giving

and hence

Of course n~2 appears in the classic unweighted estimates for polynomials, such as,

with C =1=C(n.,Pn) •

In the sequel, we will use the following notation:

• :Fwill be used to denote the class of Freud weights; £ will c' mote the class of Erdos weights and £XP the class of exponential weights.

Next, we state the Jackson theorem that can be achieved for the classes of weights mentioned above, namely, :F,£ and £XP. Recall that we use che

En

[fJ

p:= inf

11(1 -

P)

WilL

(:R) .

PeITn

p-notation

Theorem 20 . Let 0

<

p

:s;

00, r 2: 1 and W E £ or F or £XP on the

interval I. Let f~V E Lp(1) and if p = 00, let f be continuous with f~V

vanishing at the endpoints of I. Then

(2.65)

Corollary 21 . If p 2: 1, we have

(2.66)

For (2.65) and (2.66) to make sense we need to define the modulus of continuity wr,p(I, t) and the quantity <p7' First let us note that

II

<Phll

< C,

n>

1.

n Loo{l) -

-(2.68)

Definition 22 We define the rth order symmetric difference with

incre-ment h by

D.'hf (x) :=

ta

(I)

(_l)i f (x

+

G -

j)

h)

(2.67)

One of the fundamental discoveries of Ditzian and Totik was that by letting the size of the increment hdepend onx, more specifically by replacing hbyh-/1 - X2, they could achieve elegant characterizations of the condition

in terms of the structural properties of

f.

Their modulus for [-1,1] has the

form

and they showed',nat if 0<O! <r, (2.68) is equivalent to

W;,p (I,t) =0(to), t -> 0':".

1= (-1,1) orlRand the Lp norm has the form

Wr,p (1,t) =sup

IIW

(x) f:lh.~,(x)f

(x)il'L

(I)

+

inf

IIFV

(f - P)IIL (J) •

O<h::;t , p h PEPr-l P •t

---

main part tail

(2.69)

Wr,p (1,t)

==

O.

'We see that there are 3 undefined quantities: The function (pt and the

in-tervals h,Jt. Before we define them, we need to account for the "tail" part

of (2.69). We know that PnW is small outside [-an, an] and so can only

approximate inside the interval. The tail part is designed1;0take account of

this. vVeneed an inf over polynomials (Jidegree:Sr - 1in the tail to ensure that if

f

is itself a polynomial of degree j;r - 1, we have

The function (Pt is a suitable replacement for' the factor

v'f=""":2l'

in the

Ditzian-Totik modulus, and will depend on the class of weights. If(Pt

¥

1,

it reflects an improvement in the degree of approximation towards the end-points of a certain interval. To make things more explicit, we distinguish between (-1,1) and JR.:

(I) Freud and Erdos weights on lR.

For the classes of weightsE, :F, Q (x) grows faster than Ix

I'" ,

O! '>1, and

1

angrows slower than n.;;. So

an

- -+ 0, n-+ 00.

n

It turns out that we need something like an, where n. is evaluated at the inverse of 'ie function n -+ ~ to describe: the interval h.Accordingly, we introduce the decreasing function of t,

The essential feature of (J"here is that

For :F,we define

1>t (x)

==

1; h :=[-(J" (h), (J(h)]; J,:=lR\h.

For

c,

we define

and for 0 <h S;t,

h:=[-(J" (2t) ,(J" (2t)]; Jt:=

1R\

[-(J" (4t) ,(J" (4t)].

Clearly the choice of <Pt (x)

==

1 for :F does not give any improvement in the degree of approximation, while in the

c

case, the choice of 1>t reflects

an improvement in approximatior, towards the endpoints of the interval [-IT (t) .a(t)].

(II) Exponential 'Weights on (-1,1) .

An essential simplification in this case is that an -> 1, n _, 00, so we could actually replace the ~ in our Jackson theorem (2.65) by ~. This also means that we do not need the function IT(t), and instead can use just. al.

e

So here we choose

and for 0

<

h ::;t

Ii.:=

[-a.l., a.l.];

Jt:= [-1.1] \

[-a.l., a.l.] .

u u « «

Remark 4 For all three classes of weights (E,F and EXP), the modulus

does have the usual properties

6.-'r,p(j,2t) ::;Cwr,p (1,t)

and for p;::: 1,

W

(f

t)

<

C tT

Ilf(r)W<'Prll

.

r,p , - IiLp(1)

For the Freud weights, Theorem (2.7) is due to Z. Ditzian and D.S.

for exponential weights, the result is due to D.S. Lubinsky [38]. The unifying feature here is that the method of proof throughout is the unknown approx-imator's method and it works simultaneously in allLp spaces, 0<P :500.

In conclusion, we present some of the details of the unknown approxirna-tor's method in the weighted setting:

First let us recall that in approximating fW by PnH~we can only work on

[-an, an] asPn VVis very small outside [-an, an] and'JO cannot approximate.

So we partition this interval as

-an

=

Yo <YI <Y2 < ... <Yn+1

=

an

and on each interval[Yi' YJ+l) approximate

f

by a polynomialPJ of degree'S:r.

According to Whitney's theorem, we can choosePj to approximate

f

with an

error involving a suitable rth order modulus evaluated at Yj+l - Yj. Thn=;L

is really the size of(Yj+1 - Yj) that controls the degree of approximation. For

Freud weights, we can choose the Yi equally spaced, but for Erdos weights

and exponential weights on [-1.1] ,one has to choose theYJ in such a way

that uniformly inj andn

a.; (

Y)+l - YJ '" - (p~ Yj).

n n

Then one obtains a piecewise polynomial approximation to

f

n n-l

f

(x) ~ LPjX[Yj,Yj+l) =Po

+

L (PHI - Pj) X[YJ+l,an),

J=O j=O

Next, approximate X[Yj+1,Un) by a polynomial Rj of degree S n. The type of

<stimate needed here is the following: For11. 2: G and T E [-an, an] there exist polynomials

Rn,T

of degreej;Ln.such that for allx E1= (-1,1) or ~,

where G, G1•L =1= 0, G1,L(11., T,x). Moreover, 99 can be replaced b, .ny

positive integer. The estimate is difficult because W(7) may be very small

for 7close to an. In the Freud case, the <P

==

1, but in the other cases, it

gets smaller as7approaches an and this is really where one needs the Yj to